Roll-diameter dependence in Rayleigh convection and its effect upon the heat flux

AbstractConvection from an isolated heat source in a chamber has been previously studied numerically, experimentally and analytically. These have not covered long time spans for wide ranges of Rayleigh number Ra and Prandtl number Pr. Numerical calculations of constant viscosity convection partially fill the gap in the ranges $\mathit{Ra}= 1{0}^{3} {{\unicode{x2013}}}1{0}^{6} $ and $\mathit{Pr}= 1, 10, 100, 1000$ and $\infty $. Calculations begin with cold fluid everywhere and localized hot temperature at the centre of the bottom of a square two-dimensional chamber. For $\mathit{Ra}\gt 20\hspace{0.167em} 000$, temperature increases above the hot bottom and forms a rising plume head. The head has small internal recirculation and minor outward conduction of heat during ascent. The head approaches the top, flattens, splits and the two remnants are swept to the sidewalls and diffused away. The maximum velocity and the top centre heat flux climb to maxima during head ascent and then adjust toward constant values. Two steady cells are separated by a vertical thermal conduit. This sequence is followed for every value of $Pr$ number, although lower Pr convection lags in time. For $\mathit{Ra}\lt 20\hspace{0.167em} 000$ there is no plume head, and no streamfunction and heat flux maxima with time. For sufficiently large Ra and all values of Pr, an oscillation develops at roughly $t= 0. 2$, with the two cells alternately strengthening and weakening. This changes to a steady flow with two unequal cells that at roughly $t= 0. 5$ develops a second oscillation.

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A study of Bénard convection with and without rotation

Journal of Fluid Mechanics ◽

10.1017/s0022112069001674 ◽

1969 ◽

Vol 36 (2) ◽

pp. 309-335 ◽

Cited By ~ 433

Author(s):

H. T. Rossby

Keyword(s):

Boundary Layer ◽

Heat Flux ◽

Rayleigh Number ◽

Prandtl Number ◽

Linear Theory ◽

Silicone Oil ◽

Taylor Number ◽

Subcritical Instability ◽

Two Fluids ◽

Wide Range

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

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Natural Convection in a Micro Size Horizontal Cylindrical Annulus With Periodic Volumetric Heat Flux

ASME 2009 7th International Conference on Nanochannels, Microchannels and Minichannels ◽

10.1115/icnmm2009-82162 ◽

2009 ◽

Author(s):

Kamyar Mansour

Keyword(s):

Heat Flux ◽

Natural Convection ◽

Rayleigh Number ◽

Viscous Fluid ◽

Two Dimensional ◽

Dimensional Problem ◽

Similarity Parameter ◽

Symbolic Calculation ◽

Cylindrical Annulus ◽

Gap Ratio

We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) Horizontal Cylindrical annulus filled with viscous fluid and periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of four, and Rayleigh number and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.

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The effects of thermal conditions on the cell sizes of two-dimensional convection

Journal of Fluid Mechanics ◽

10.1017/s0022112094003022 ◽

1994 ◽

Vol 281 ◽

pp. 33-50 ◽

Cited By ~ 16

Author(s):

Masaki Ishiwatari ◽

Shin-Ichi Takehiro ◽

Yoshi-Yuki Hayashi

Keyword(s):

Heat Flux ◽

Lower Boundary ◽

Thermal Conditions ◽

Heat Fluxes ◽

Internal Cooling ◽

Two Dimensional ◽

Numerical Integrations ◽

The Difference ◽

Convective Cells ◽

Upper Boundary

The effects of thermal conditions on the patterns of two-dimensional Boussinesq convection are studied by numerical integration. The adopted thermal conditions are (i) the heat fluxes through both upper and lower boundaries are fixed, (ii) the same as (i) but with internal cooling, (iii) the temperature on the lower boundary and the heat flux through the upper boundary are fixed, (iv) the same as (iii) but with internal cooling, and (v) the temperatures on both upper and lower boundaries are fixed. The numerical integrations are performed with Ra = 104 and Pr = 1 over the region whose horizontal and vertical lengths are 8 and 1, respectively.The results confirm that convective cells with the larger horizontal sizes tend to form under the conditions where the temperature is not fixed on any boundaries. Regardless of the existence of internal cooling, one pair of cells spreading all over the region forms in the equilibrium states. On the other hand, three pairs of cells form and remain when the temperature on at least one boundary is fixed. The formation of single pairs of cells appearing under the fixed heat flux conditions shows different features with and without internal cooling. The difference emerges as the appearance of a phase change, whose existence can be suggested by the weak nonlinear equation derived by Chapman & Proctor (1980).

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Finite Prandtl number convection with nearly insulating boundaries

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700004871 ◽

1981 ◽

Vol 24 (3) ◽

pp. 339-347

Author(s):

N. Riahi

Keyword(s):

Heat Flux ◽

Prandtl Number ◽

Maximum Amount ◽

Two Dimensional ◽

Thermal Conductivities

Arbitrary Prandtl number convection in a layer with nearly insulating boundaries is investigated. For B1/3 > 3.95P (P is the Prandtl number and B being the ratio between the thermal conductivities of the boundary and of the fluid) two-dimensional rolls are stable. The heat transported by the stable rolls reaches its peak at a critical P = B1/3/5.77 beyond which the heat flux decreases with increasing P. For B1/3 = 3.95P rolls become unstable to disturbances in the form of rolls oriented at a right angle to the original rolls. For B1/3 > 3.95P square pattern convection represents the preferred stable convection. The stable square pattern transports the maximum amount of heat at a critical P = B1/3/3.7.

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The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1981.0167 ◽

1981 ◽

Vol 378 (1775) ◽

pp. 539-566 ◽

Cited By ~ 8

Keyword(s):

Rayleigh Number ◽

Prandtl Number ◽

Finite Amplitude ◽

Two Dimensional ◽

Bénard Convection ◽

Benard Convection ◽

Side Walls ◽

Rayleigh Benard Convection ◽

Rayleigh Numbers ◽

Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

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NATURAL CONVECTION IN HIGH ASPECT RATIO THREE-DIMENSIONAL ENCLOSURES WITH UNIFORM HEAT FLUX ON THE HEATED WALL

Revista de Engenharia Térmica ◽

10.5380/reterm.v3i2.3530 ◽

2004 ◽

Vol 3 (2) ◽

pp. 100

Author(s):

T. Dias Jr. ◽

L. F. Milanez

Keyword(s):

Heat Flux ◽

Nusselt Number ◽

Natural Convection ◽

Rayleigh Number ◽

Aspect Ratio ◽

Prandtl Number ◽

High Aspect Ratio ◽

Three Dimensional ◽

Uniform Heat Flux ◽

Uniform Heat

In this work, the laminar natural convection in high aspect ratio three-dimensional enclosures has been numerically studied. The enclosures studied here were heated with uniform heat flux on a vertical wall and cooled at constant temperature on the opposite wall. The remaining walls were considered adiabatic. Fluid properties were assumed constant except for the density change with temperature on the buoyancy term. The governing equations were solved using the finite volumes method and the dimensionless form of these equations has the Prandtl number and the modified Rayleigh number as parameters. The influences of the Rayleigh number and of the cavity aspect ratio on the Nusselt number, for a Prandtl number of 0.7, were analyzed. Results were obtained for values of the modified Rayleigh number up to 106 and for aspect ratios ranging from 1 to 20. The results were compared with two-dimensional results available in the literature and the variation of the average Nusselt number with the parameters studied were discussed.

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Horizontal convection is non-turbulent

Journal of Fluid Mechanics ◽

10.1017/s0022112002001313 ◽

2002 ◽

Vol 466 ◽

pp. 205-214 ◽

Cited By ~ 91

Author(s):

F. PAPARELLA ◽

W. R. YOUNG

Keyword(s):

Thermal Diffusivity ◽

Energy Dissipation ◽

Rayleigh Number ◽

Prandtl Number ◽

Numerical Solutions ◽

Inviscid Limit ◽

Two Dimensional ◽

Uniform Temperature ◽

Horizontal Convection ◽

Boussinesq Fluid

Consider the problem of horizontal convection: a Boussinesq fluid, forced by applying a non-uniform temperature at its top surface, with all other boundaries insulating. We prove that if the viscosity, ν, and thermal diffusivity, κ, are lowered to zero, with σ ≡ ν/κ fixed, then the energy dissipation per unit mass, κ, also vanishes in this limit. Numerical solutions of the two-dimensional case show that despite this anti-turbulence theorem, horizontal convection exhibits a transition to eddying flow, provided that the Rayleigh number is sufficiently high, or the Prandtl number σ sufficiently small. We speculate that horizontal convection is an example of a flow with a large number of active modes which is nonetheless not ‘truly turbulent’ because ε→0 in the inviscid limit.

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Boundary-layer solutions of single-mode convection equations

Journal of Fluid Mechanics ◽

10.1017/s0022112081002607 ◽

1981 ◽

Vol 102 ◽

pp. 211-219 ◽

Cited By ~ 2

Author(s):

N. Riahi

Keyword(s):

Boundary Layer ◽

Heat Flux ◽

Rayleigh Number ◽

Prandtl Number ◽

Thermal Convection ◽

Single Mode ◽

Layer Method ◽

Boundary Layer Method ◽

Inertial Terms ◽

Large Rayleigh Number

Nonlinear thermal convection between two stress-free horizontal boundaries is studied using the modal equations for cellular convection. Assuming a large Rayleigh number R the boundary-layer method is used for different ranges of the Prandtl number σ. The heat flux F is determined for the values of the horizontal wavenumber a which maximizes F. For a large Prandtl number, σ [Gt ] R⅙(log R)−1, inertial terms are insignificant, a is either of order one (for $\sigma \geqslant R^{\frac{2}{3}}$) or proportional to $R^{\frac{1}{3}}\sigma^{-\frac{1}{2}}$ (for $\sigma \ll R^{\frac{2}{3}}$) and F is proportional to $R^{\frac{1}{3}}$. For a moderate Prandtl number, \[ (R^{-1}\log R)^{\frac{1}{9}} \ll \sigma \ll R^{\frac{1}{6}}(\log R)^{-1}, \] inertial terms first become significant in an inertial layer adjacent to the viscous buoyancy-dominated interior, and a and F are proportional to R¼ and \[ R^{\frac{3}{10}}\sigma^{\frac{1}{5}} (\log\sigma R^{\frac{1}{4}})^{\frac{1}{10}}, \] respectively. For a small Prandtl number, $R^{-1} \ll \sigma \ll (R^{-1} \log R)^{\frac{1}{9}}$, inertial terms are significant both in the interior and the boundary layers, and a and F are proportional to ($R \sigma)^{\frac{9}{32}} (\log R\sigma)^{-\frac{1}{32}}$ and ($R \sigma)^{\frac{5}{16}} (\log R \sigma)^{\frac{3}{16}}$, respectively.

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NATURAL CONVECTION IN HIGH ASPECT RATIO THREE-DIMENSIONAL ENCLOSURES WITH UNIFORM HEAT FLUX ON THE HEATED WALL

Revista de Engenharia Térmica ◽

10.5380/ret.v3i2.3530 ◽

2004 ◽

Vol 3 (2) ◽

Author(s):

T. Dias Jr. ◽

L. F. Milanez

Keyword(s):

Heat Flux ◽

Nusselt Number ◽

Natural Convection ◽

Rayleigh Number ◽

Aspect Ratio ◽

Prandtl Number ◽

High Aspect Ratio ◽

Three Dimensional ◽

Uniform Heat Flux ◽

Uniform Heat

In this work, the laminar natural convection in high aspect ratio three-dimensional enclosures has been numerically studied. The enclosures studied here were heated with uniform heat flux on a vertical wall and cooled at constant temperature on the opposite wall. The remaining walls were considered adiabatic. Fluid properties were assumed constant except for the density change with temperature on the buoyancy term. The governing equations were solved using the finite volumes method and the dimensionless form of these equations has the Prandtl number and the modified Rayleigh number as parameters. The influences of the Rayleigh number and of the cavity aspect ratio on the Nusselt number, for a Prandtl number of 0.7, were analyzed. Results were obtained for values of the modified Rayleigh number up to 106 and for aspect ratios ranging from 1 to 20. The results were compared with two-dimensional results available in the literature and the variation of the average Nusselt number with the parameters studied were discussed.

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